This is not a generalized linear model (well, technically it is), it is just a plain old linear model!
You can define a dummy variable for each of the groups (except one "reference group") You can then build your individuals-variables matrix as in any linear model, and solve by least-squares to estimate the parameters.
EDIT: I am not sure about what exactly counts as a "mixed model". Many of those names are used for historical reasons, as the general formulation of the linear model has only been around for about half a century (It is confusing, but please don't mistake "general linear model" with "generalized linear model". The former refers to any model than can be written as $Y=AX+\epsilon$ while the latter also allows for some transformations that make it suitable for other problems like classification (logistic regression is probably the most famous case of a generalized linear model)
Another reason why those specific names exist is clarity in description, as, although mathematically the model you describe is just "multivariant regression", from a "functional" point of view, it's quite different (those "dummy variables" are just a trick we use to make it fit into the linear model form, but we don't often think of cateogrical variables as their dummy representation)
Finally, I guess pretending we are working with a set of different models, rather than a general one, makes everyone else think we are smarter!